The Newton Polygon of Plane Curves with Many Rational Points

Abstract

This curve has the points (1 : 0 : 0) and (0 : 1 : 0) at infinity over any field. The affine equation is XY +Y +X = 0. The origin is a point of this curve. If (x, y) ∈ F8 is a point of this curve with nonzero coordinates, then x = 1. So 0 = xy + y + x = xy + xy + x = x[(xy) + (xy) + 1]. Let t = xy. Then t + t+ 1 = 0. So the Klein quartic has 3.7 = 21 rational points over F8 with nonzero coordinates. ∗Department of Mathematics and Computing Science, Technical University of Eindhoven , P.O. Box 513, 5600 MB Eindhoven, The Netherlands.